Robust controller for electro-mechanical actuators employing sliding and second control modes

ABSTRACT

An improved technique for controlling an electro-mechanical actuator combines a sliding mode of control with a second mode of control. An error signal is generated based on the difference between an input position signal and a feedback position signal. When the error signal is above a predetermined threshold, the actuator is controlled in the sliding control mode. When the error signal is below the predetermined threshold, the actuator is controlled in the second control mode. The combination of the sliding control mode with the second control mode yields a robust controller that can tolerate large parameter variations and uncertainties without sacrificing precise steady state tracking.

BACKGROUND

Electro-mechanical actuators are used on airborne vehicles and guidedprojectiles to establish and maintain the positions ofposition-controlled elements (PCEs), such as fins, flaps and otherflight control surfaces. Mechanical power is generated by a motor withinan electro-mechanical actuator and coupled to a PCE via a mechanicaldrive linkage. Control of an actuator is typically managed by a controlcircuit, or “controller,” which is responsible for accuratelypositioning the PCE in response to a positioning command. Thepositioning command may be generated by a navigation system, forexample, which is responsible for moving the airborne vehicle orprojectile along a desired flight path. In some examples, thepositioning command is expressed as an angle, which corresponds to adesired angular position of the PCE.

In a typical arrangement, the controller for an electro-mechanicalactuator receives a position command signal as well as a positionfeedback signal indicating the actual position of the PCE. The positionfeedback signal may be provided from a Hall-effect sensor within themotor of the electro-mechanical actuator. The controller processes theposition command signal and the feedback signal to generate a controlsignal, which drives the actuator's motor. The controller can thuscontrol the electro-mechanical actuator to establish and maintain theactual position of the PCE at the desired position prescribed by theposition command signal.

One general class of controller for electro-mechanical actuators is theproportional-integral-derivative, or “PID,” controller. The PIDcontroller allows a designer to specify parameters of separateproportional, integral, and derivative blocks. Designers can place polesand zeroes in the controller's transfer function to compensate fordynamics of the motor and the electro-mechanical actuator, forestablishing stability and desired response characteristics. The use ofPID controllers in connection with motors is discussed, for example, byR. Krishnan in “Electric Motor Drives Modeling, Analysis, and Control,”Prentice Hall, N.J., 2001.

Other classes of controllers for electro-mechanical actuators includeoptimal and adaptive control schemes. Optimal controllers show anadvantage over PID controllers where the design goal is to provide anoptimized control effort within an assumed range of parametervariations. Adaptive controllers, such as gain-scheduled controllers,can vary their parameters to adapt to changes in their operatingenvironments.

SUMMARY

The control of electro-mechanical actuators in airborne applicationspresents particular challenges. For example, the environmentaltemperature in which the actuators operate typically varies over a widerange, causing temperature-dependent changes in load characteristics. Inaddition, manufacturing tolerances of motor parameters and transmissionefficiency vary widely, such that motor characteristics can beconsiderably different from one unit to the next. Also, whereelectro-mechanical actuators are powered from batteries, batteryvoltages can be uncertain with large manufacturing tolerances andbattery voltages may change substantially with temperature. Further,flight duty cycle, i.e., the torque required to move a position controlelement, typically changes substantially and non-linearly as airspeedchanges. These factors present difficult challenges in controllingelectro-mechanical actuators for airborne applications.

Unfortunately, PID controllers, optimal controllers, and adaptivecontrollers tend to be ill-suited for operation involving such variableand non-linear characteristics. Although PID controllers can typicallybe tuned to perform well under one set of conditions, they tend to beless well-suited when conditions change. Similarly, optimal controllersare typically tuned for a narrow band of parameter variations, but theirperformance typically degrades rapidly outside that band. Performance ofoptimal controllers also degrades in the face of high frequencyperturbed dynamics. Adaptive controllers can usually be stabilized overa wide range of operating parameters; however, such stability istypically achieved by substantially increasing the order of suchcontrollers, which results in complex designs with very high latency.

We have recognized that another type of controller is well suited incertain respects for the challenges at hand. This type of controller,known as a sliding mode controller, can be designed to behaveconsistently in the face of large parameter variations andnon-linearities. Sliding mode controllers operate by generating atime-varying sliding function, s(t), where s(t)=0 defines an invariantsliding surface in a phase plane. The sliding function is calculated asa weighted sum of a difference signal and its derivative(s), where thedifference signal is the difference between a desired value of theoutput of interest and a feedback value. Operation of feedback tends todrive the output state trajectory to the sliding surface. Once thesliding surface is reached, feedback further tends to drive the outputof interest to the desired value by driving the output state trajectoryalong the sliding surface in the phase plane with first-order settlingcharacteristics. The overall system being controlled may have a highorder, but the sliding function is constrained such that it behaves as afirst order system, greatly simplifying control. The theoretical basisfor sliding mode control is explained, for example, in Applied NonlinearControl, by Slotine and Li (Slotine, J. J. E., and W. Li, AppliedNonlinear Control, Prentice-Hall (1991)).

Although sliding mode control confers distinct advantages in variableand uncertain environments, it tends to suffer from a significantdrawback—sliding mode control tends to cause chattering in the vicinityof the sliding surface, resulting in high frequency and high speedinstability. Chattering results from switching between opposing controlmagnitudes at the sliding surface. Although sliding mode control canaccount for the presence of modeling imprecision and of parameteruncertainties, the problem of chattering makes sliding controllers aless-than-ideal solution for controlling actuators in airborneapplications.

In contrast with these prior approaches, an improved technique forcontrolling an electro-mechanical actuator combines a sliding mode ofcontrol with a second mode of control that is not susceptible tochattering. An error signal is generated based on the difference betweenan input position signal and a feedback position signal. When the errorsignal is above a predetermined threshold, the actuator is controlled inthe sliding control mode. When the error signal is below thepredetermined threshold, the actuator is controlled in the secondcontrol mode. Sliding mode control brings the actuator close to aninvariant sliding surface, even in the face of wide variations anduncertainties in system parameters, while the second-control mode takesover as the invariant sliding surface is approached, to promote precisetracking without chattering.

With the improved technique, chattering may still arise during slidingmode control, but sliding mode control is generally applied only whenthe output trajectory of the electro-mechanical actuator in the phaseplane is far from the sliding surface, e.g., when the position of theelectro-mechanical actuator is changing, such as when new inputs arereceived and when responding to perturbations. Chattering typically doesnot arise once a steady-state value is approached, as control in thevicinity of steady state is maintained by the second control mode.

According to one variant, the effects of chattering are further reducedby establishing a boundary layer around the sliding surface andproviding continuous gain across the boundary layer, thereby eliminatingthe discontinuity in gain across the sliding surface that wouldotherwise be present. Chattering can thus be substantially reduced oreliminated in most if not all cases.

In an example, the second control mode is PID control; however, this isnot required. Alternatively, the second control mode may be realizedwith Kalman control, H2 control, or H-infinity control, and so forth,for example.

Certain embodiments are directed to a method of controlling anelectro-mechanical actuator. The method includes receiving a firstsignal indicating a desired position of the electro-mechanical actuatorand receiving a second signal indicating an actual position of theelectro-mechanical actuator. An error signal is calculated based on thefirst signal and the second signal. The method further includescontrolling the position of the electro-mechanical actuator in a slidingcontrol mode when the error signal is above a predetermined thresholdand controlling the position of the electro-mechanical actuator in asecond control mode when the error signal is below the predeterminedthreshold.

Other embodiments are directed to a control circuit for controlling anelectro-mechanical actuator. The control circuit includes a sliding modecontroller configured to generate a sliding mode control signal and asecond controller configured to generate a second mode control signal.The control circuit further includes an error circuit configured togenerate an error signal based on a difference between an input signalindicative of a desired position of the electro-mechanical actuator anda feedback signal indicative of an actual position of theelectro-mechanical actuator. The control circuit still further includesa selector circuit coupled to the sliding mode controller, the secondcontroller, and the error circuit. The selector circuit is configured(i) to select the sliding mode control signal to control theelectro-mechanical actuator when the error signal is above apredetermined threshold and (ii) to select the second mode controlsignal to control the electro-mechanical actuator when the error signalis below the predetermined threshold.

Other embodiments are directed to computerized apparatus and computerprogram products. Some embodiments involve activity that is performed ata single location, while other embodiments involve activity that isdistributed over a computerized environment (e.g., over a network).

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The foregoing and other features and advantages will be apparent fromthe following description of particular embodiments of the invention, asillustrated in the accompanying drawings, in which like referencecharacters refer to the same parts throughout the different views. Inthe accompanying drawings,

FIG. 1 is a block diagram of an example apparatus in which the positionof an electro-mechanical actuator is controlled according to theimprovements hereof;

FIG. 2 is a block diagram of the control circuit of FIG. 1, including asliding mode controller and a second controller;

FIG. 3 is a block diagram of an example model of the brushless DC motorshown in FIG. 1;

FIG. 4 is a block diagram showing portions of the example sliding modecontroller of FIG. 2;

FIG. 5 is a block diagram showing additional portions of the examplesliding mode controller of FIG. 2;

FIG. 6 is a phase plot of a sliding function of the example slidingcontroller of FIG. 2, which illustrates the problem of chattering whenthere is no boundary layer;

FIG. 7 is a phase plot of the sliding function of the example slidingcontroller of FIG. 2, which shows a reduction in chattering through theuse of a boundary layer around the sliding surface through which gain ofthe sliding controller is transitioned to reduce discontinuities;

FIG. 8 is a block diagram of an example second controller of FIG. 2; and

FIG. 9 is a flowchart showing an example process for controlling anelectro-mechanical actuator using a sliding control mode and a secondcontrol mode according to improvements hereof.

DETAILED DESCRIPTION OF THE INVENTION

Embodiments of the invention will now be described. It is understoodthat such embodiments are provided by way of example to illustratevarious features and principles of the invention, and that the inventionhereof is broader than the specific example embodiments disclosed.

An improved technique for controlling an electro-mechanical actuatorcombines a sliding mode of control with a second mode of control. Anerror signal is generated based on the difference between an inputposition signal and a feedback position signal. When the error signal isabove a predetermined threshold, the actuator is controlled in thesliding control mode. When the error signal is below the predeterminedthreshold, the actuator is controlled in the second control mode. Thecombination of the sliding control mode with the second control modeyields a robust controller that can tolerate large parameter variationsand uncertainties without sacrificing precise steady state tracking.

FIG. 1 shows an example apparatus 100 in which the position of anelectro-mechanical actuator 140 is controlled using the improvedtechnique hereof. The apparatus 100 is seen to include a control circuit110, which combines a sliding control mode and a second control modewhich are selected based on a predetermined threshold 114. An inputsignal 112 received, for example, from a guidance system, indicates adesired position of the electro-mechanical actuator 140. The controlcircuit 110 produces an output in the form of a control signal 116. Thecontrol signal 116 drives a pulsewidth modulator 120, which is coupledto a bridge 130. A power source, such as a battery 132, provides powerto the bridge 130. A measuring circuit, such as an analog-to-digitalconverter 134, is coupled to the battery 132, to measure the voltageacross the battery 132 and to produce a battery voltage signal 136. Thebridge 130 is coupled to a brushless DC motor 142 of theelectro-mechanical actuator 140. The brushless DC motor 142 has a shaft144, which is coupled to a position control element, such as a flap 150.Other mechanical linkages, such as gears and other couplings, may beprovided. In the example shown, the angle of the flap 150 is varied byvarying the angle of the shaft 144. It is understood that other types ofposition control elements can be provided that can be moved in rotation,translation, or in any other manner, as appropriate for their purposes,and that such position control elements may be controlled in a similarmanner to that shown. A sensor 146 is provided, e.g., within theelectro-mechanical actuator 140, to measure the rotational position ofthe shaft 144. In an example, the sensor 146 is a Hall-effect sensor.The sensor 146 generates a feedback signal 148, which is provided to thecontrol circuit 110. Other types of sensors may be used, such as opticalencoders, magnetic encoders, potentiometers, and resolvers, formeasuring the angular position of the shaft 144 of the brushless DCmotor 142 or position of the flap 150.

In operation, the control circuit 110 generates the control signal 116based on a difference between the desired position 112 and the actualposition as indicated by the feedback signal 148. The pulsewidthmodulator 120 receives the control signal 116 and generates outputsignals for driving the bridge 130. In an example, the output signals ofthe pulsewidth modulator 120 are provided in the form of rectangularwaveforms having constant frequency but variable pulsewidth. As thecontrol signal 116 increases, the pulsewidth modulator 120 produceslonger pulsewidths. As the control signal 116 decreases, the pulsewidthmodulator 120 produces shorter pulsewidths. In an example, the bridge130 is an H-bridge that includes four switching elements (e.g.,transistors) arranged in an “H” configuration. The bridge 130 has twooutputs, which are coupled to a set of windings of the brushless DCmotor 142. Depending on the signals from the pulsewidth modulator 120,the voltage applied to the windings of the brushless DC motor 142 isequal to either zero volts, the voltage of the battery 132, or thenegative of the voltage of the battery 132. The brushless DC motor 142responds to pulsed voltage applied to its windings by rotating the shaft144 in a controlled manner, either clockwise or counterclockwise. As theshaft 144 rotates, the flap 150 rises or lowers. The sensor 146 measuresthe rotational position of the shaft 144 and reports the position to thecontrol circuit 110 via the feedback signal 148. As the rotation of theshaft 144 corresponds directly to the position of the flap 150, thefeedback signal 148 provides an accurate measure of the actualrotational position of the flap 150.

The control circuit 110 controls the position of the flap 150 usingfeedback. For example, the control circuit 110 receives the feedbacksignal 148, compares it with the input signal 112, and varies the levelof the control signal 116 to drive the feedback signal 148 to the valuethat corresponds to the desired position 112. In controlling theposition of the flap 150, the control circuit 110 alternately applies asliding control mode and a second control mode. The control circuit 110applies the sliding control mode when an error signal based on thedesired position 112 and the feedback position 148 exceeds thepredetermined threshold 114 and applies the second control mode when theerror signal is less than the predetermined threshold 114. The controlcircuit 110 thus benefits from the ability of the sliding control modeto bring the position of the flap 150 close to the invariant slidingsurface (i.e., close to steady state), even when faced with widevariations and uncertainties in operating parameters, as it alsobenefits from the ability of the second control mode to provide accuratesteady-state tracking without chattering.

The value of the predetermined threshold 114 can be established asappropriate for the particular target application by trial and error. Inan example, the threshold 114 is set at 1% of the full-scale range ofthe control signals 242 and 252.

FIG. 2 shows the control circuit 110 in additional detail. As shown, thecontrol circuit 110 includes a first converter 210, a second converter220, an error circuit 230, a sliding mode controller 240, a secondcontroller 250, a selector 260, and a source voltage normalizationcircuit 270. The first converter 210 converts the desired positionsignal 112 into units of radians (or some other suitable units). In anexample, where the desired position signal 112 is received in units ofdegrees, the first converter 210 converts the desired position signal112 into a first converted signal 212 (Θ_(d)), expressed in units ofradians, by multiplying by π/180. The second converter 220 converts thefeedback signal 148 into a second converted signal 222 (Θ), which isexpressed in the same units as the first converted signal 212 (e.g.,radians). In addition to performing unit conversion, the first andsecond converters 210 and 220 may perform other corrections, as needed,such as to correct for variable gear ratios or other knowncharacteristics the motor 142, sensor 146, flap 150, and any linkagesbetween the motor 142 and the flap 150.

The error circuit 230 includes a difference circuit 232 and anormalizing circuit 234. The difference circuit 232 receives the firstand second converted signals 212 and 222 and generates a differencesignal 236 (Diff) equal to the difference between these signals. Thedifference signal 236 thus corresponds to the difference between thedesired position signal 112 and the feedback position signal 148. Thenormalizing circuit 234 receives the difference signal 236 as well asthe first converted signal 212 and generates an error signal 238 (e.g.,a normalized error signal, E_(Norm)). In an example, the error signal238 is computed as the absolute value of the quotient of the differencesignal 236 divided by the first converted signal 212. The error signal238 is thus expressed as a number greater than or equal to zero, whosevalue scales relative to the magnitude of the first converted signal212, i.e., in relation to the magnitude of the desired position signal112.

The sliding mode controller 240 receives the first and second convertedsignals 212 and 222 as input, as well as the difference signal 236. Thesliding mode controller 240 operates in response to its inputs togenerate a sliding mode control signal 242. The second controller 250also receives the difference signal 236 and operates in response to thedifference signal 236 to generate a second control signal 252. Dependingon the particular design of the second controller 250, the secondcontroller 250 may receive additional input signals.

The selector 260 receives the sliding mode control signal 242 and thesecond control signal 252 and selects between them to produce an outputsignal 262. The operation of the selector 260 is based on a comparisonof the error signal 238 with the predetermined threshold 114. If theerror signal 238 is greater than or equal to the threshold 114, theselector 260 selects the sliding mode control signal 242, i.e., theselector 260 provides the sliding mode control signal 242 as the output262. If the error signal 238 is less than the threshold 114, theselector 260 selects the second control signal 252 as the output signal262.

The source voltage normalization circuit 270 adjusts the output signal262 of the selector 260 to adjust for variations in voltage from thebattery 132. For example, the source voltage normalization circuit 270divides the output signal 262 by the battery voltage signal 136, toproduce the control signal 116, i.e., the output of the control circuit110.

Compensation for battery voltage is based on the observation that theopen-loop gain of the apparatus 100 varies in proportion to the voltageof the battery 132. Battery voltage can vary substantially withmanufacturing tolerances, changes in temperature, and with use. Thegreater the voltage on the battery 132, the greater the voltage appliedto the motor 142 and the greater the motor's slew rate response.Normalizing the output signal 262, i.e., by dividing the output signal262 by the voltage of the battery 132, thus has the effect ofcompensating for changes in open-loop gain that occur as a result ofbattery voltage changes. In some examples, the measurement circuit 134measures the voltage of the battery 132 at a high rate, such that thebattery voltage signal 136 tracks changes in the voltage of the battery132 as they occur and the source voltage normalization circuit 270corrects for changes in battery voltage in real time.

Preferably, the control circuit 110 is implemented in digital form,where the components of the control circuit 110 operate synchronously inaccordance with a clock. The desired position signal 112 and thepredetermined threshold 114 are digital values, and the feedbackposition signal 148 and the battery voltage signal 136 are discrete-timesampled digital signals.

The control circuit 110 can be constructed in any suitable way. Forexample, the control circuit 110 can be implemented as anApplication-Specific Integrated Circuit (ASIC), a Field-ProgrammableGate Array (FPGA), with one or more DSP (Digital Signal Processing)units, microprocessors, computers, and/or any combination of the above,for example. It is understood that the identified components of thecontrol circuit 110 need not be physically distinct structures. Forexample, any of the structures shown in FIG. 2 can be implemented incode residing in memory of a computing device and run by one or moreprocessors. Indeed, some of the structures shown (e.g., the converters210 and 220, the selector 260, and the source voltage normalizationcircuit 270 can be implemented in as little as a few lines of code). Thecode can be stored in ROM or in RAM, in the form of software, firmware,or custom hardware. Such code executed by one or more processors orcustom hardware forms the specialized circuit components shown in FIG.2.

FIG. 3 shows a model 300 that represents the behavior of the brushlessDC motor 142. As the brushless DC motor 142 is typically the mostvariable and unpredictable element in the apparatus 100, the model 300provides a basis for selecting design parameters of the slidingcontroller 240.

Here, the model 300 is seen to receive an input signal 310 (V_(M)) ofthe brushless DC motor 142, which corresponds to the pulsed voltageapplied to the brushless DC motor 142 from the bridge 130, and togenerate an output signal 330 (Θ(s)), which corresponds to the angle ofthe motor shaft 144. The model 300 is seen to include a summer 312, afirst block 314, a first gain element 316, a second gain block 318, anintegrator 320, and a second gain element 332, connected as shown.

In the model 300, L_(M) represents the motor inductance (Henrys), R_(M)represents the motor resistance (Ohms), K_(T) represents the motortorque constant (in-lbf/Amp), J_(M) represents the motor rotor inertia(in-lbf-sec²), and B_(M) represents the motor viscous dampingcoefficient (in-lbf-sec/radian). In this example, the motor currentlimit logic, cogging torque, commutation loss, torque ripple effect, andload side structural dynamics are ignored for simplicity.

In computing an overall transfer function of the form Θ(s)/V_(M), it isnoted that the third order term of the denominator can be neglectedwithout loss of model accuracy, as the electric time constant(L_(M)/R_(M)) is very small. Thus, the following transfer function forthe brushless DC motor 142 is obtained:

$\begin{matrix}{\frac{\theta (s)}{V_{m}(s)} = \frac{K_{T}\text{/}\left( {{R_{m}B_{m}} + {K_{T}K_{b}}} \right)}{{\tau \cdot s^{2}} + s}} & (1)\end{matrix}$

where τ is the mechanical time constant expressed as:

$\begin{matrix}{\tau = \frac{R_{m}J_{m}}{{R_{m}B_{m}} + {K_{T}K_{b}}}} & (2)\end{matrix}$

Thus, a differential equation of the brushless DC motor 142 isdetermined as

$\begin{matrix}{{{\tau \cdot \overset{¨}{\theta}} + \overset{.}{\theta}} = {\frac{K_{T}}{{R_{m}B_{m}} + {K_{T}K_{b}}} \cdot V_{m}}} & (3)\end{matrix}$

FIGS. 4 and 5 show an example detailed implementation of the slidingmode controller 240. The design parameters for the sliding modecontroller 240 are based in part on the model 300 of the brushless DCmotor 142. A mathematical development of the design parameters of thesliding mode controller 240 will now be described.

For realizing the sliding mode controller 240, it is noted that Equation(3) can be rearranged as follows:

$\begin{matrix}{{\overset{¨}{\theta} = {f + u_{SMC}}},{where}} & (4) \\{{f = {{- \frac{1}{\tau}}\overset{.}{\theta}}}{and}} & \left( {5a} \right) \\{u_{SMC} = {\frac{K_{T}}{R_{m}J_{m}} \cdot {V_{m}.}}} & \left( {5b} \right)\end{matrix}$

An estimated model of nominal state maybe defined as:

{tilde over (ƒ)}=A·{dot over (θ)}.  (6)

Also, the following inequality is established from the Equations (5a)and (6):

$\begin{matrix}{{{{\overset{\sim}{f} - f}} = {{{\left( {A + \frac{1}{\tau}} \right) \cdot \overset{.}{\theta}}} \leq {M(t)}}},{where}} & (7) \\{{{M(t)} = {M_{c} \cdot {\overset{.}{\theta}}}}{and}} & {8(a)} \\{M_{c} = {{{A + \frac{1}{\tau}}}.}} & \left( {8b} \right)\end{matrix}$

To track the first converted signal 212, i.e., θ(t)=θ_(d)(t), thesliding function is defined, as follows:

$\begin{matrix}{{s = {{\left( {\frac{}{t} + \lambda} \right)^{2} \cdot {\int_{0}^{t}{e\ {t}}}} = {\overset{.}{e} + {2 \cdot \lambda \cdot e} + {\lambda^{2} \cdot {\int_{0}^{t}{e\ {t}}}}}}},{with}} & (9) \\{{e(t)} = {{\theta (t)} - {{\theta_{d}(t)}.}}} & (10)\end{matrix}$

Here the θ_(d)(t) denotes the first converted signal 212, whichcorresponds to the desired position 112. The sliding function is definedabove to include an integral term to drive to zero steady-state errorscaused by torque disturbances from coupled load-side structures. Theunique solution on s(t)=0 (the sliding surface) is: e(t)=0, for all t>0,where e(t) corresponds to the difference signal 236. The problem oftracking is to ensure that Θ(t)=Θ_(d)(t) (the first converted signal212) remains on the sliding surface s(t)=0 of the phase plane.

The best estimate of equivalent control is obtained by {dot over (s)}=0.From the Equations (4) and (9),

{dot over (s)}=ƒ+u _(SMC)−{umlaut over (θ)}_(d)+2·λ·ė+λ ² ·e=0  (11)

Thus, the control estimate is derived as:

ũ _(SMC)=−{tilde over (ƒ)}+{umlaut over (θ)}_(d)−2·λ·ė−λ ² ·e  (12)

The sliding mode control law is then determined as:

u _(SMC) =ũ _(SMC) −k·sgn(s), where k=M(t)+η.  (13)

By substitution of Equations (6), (8), and (12) into the Equation (13),the control law takes the following form.

u _(SMC) =−A{dot over (θ)}+{umlaut over (θ)} _(d)−2·λ·ė−λ ² ·e−(M_(c)·|{dot over (θ)}(t)|+η)·sgn(s).  (14)

The simplified, first-order problem of keeping the scalar “s” at zero(i.e., on the sliding surface) can thus be achieved by choosing thecontrol law u_(SMC) of Equation (13) such that

$\begin{matrix}{{{\frac{1}{2}\frac{}{t}s^{2}} \leq {{- \eta} \cdot {s}}},} & (15)\end{matrix}$

where η is a strictly positive constant.

By substitution of Equations (4), (9), (10), (12), and (13) intoEquation (15), the condition is checked as follows.

$\begin{matrix}{{{\frac{1}{2}\frac{}{t}s^{2}} = {{{\left( {f - \overset{\sim}{f}} \right) \cdot s} - {k \cdot {s}}} \leq {\left( {{{f - \overset{\sim}{f}}} - k} \right) \cdot {s}}}}{and}} & \left( {16a} \right) \\{{{\left( {{{f - \overset{\sim}{f}}} - k} \right) \cdot {s}} \leq {\left( {{M(t)} - k} \right) \cdot {s}}} = {{- \eta} \cdot {{s}.}}} & \left( {16b} \right)\end{matrix}$

Therefore, it is observed that the sliding condition is satisfied.

Although the control law of Equation (14) guarantees convergence infinite time, some undesirable chattering at the sliding surface oftenoccurs due to the discontinuous nature of the SGN function. The inherentnature of chattering might be further exacerbated by the ignoredcomponents, such as torque ripple, load side structural dynamics, and soforth. The problem of chattering can be reduced or eliminated altogetherby establishing a boundary layer of thickness Φ and approximatelylinearizing the control effort when |s| falls inside of the boundarylayer. Thus,

$\begin{matrix}{{{{if}\mspace{14mu} {s}} < \Phi}\begin{matrix}{u_{SMC}^{\prime} = {{\overset{\sim}{u} - \frac{s}{\Phi}} =}} \\{{{{- A}\; \overset{.}{\theta}} + {\overset{¨}{\theta}}_{d} - {2 \cdot \lambda \cdot \overset{.}{e}} - {\lambda^{2} \cdot e} - \frac{\overset{.}{e} + {2 \cdot \lambda \cdot e} + {\lambda^{2} \cdot {\int_{0}^{t}{e\ {t}}}}}{\Phi}}}\end{matrix}} & (17) \\{{{{elseif}\mspace{14mu} {s}} \geq \Phi},{u_{SMC} = {{{- A}\; \overset{.}{\theta}} + {\overset{¨}{\theta}}_{d} - {2 \cdot \lambda \cdot \overset{.}{e}} - {\lambda^{2} \cdot e} - {\left( {{M_{c} \cdot {{\overset{.}{\theta}(t)}}} + \eta} \right) \cdot {{sgn}(s)}}}}} & (14)\end{matrix}$

It is noted that the velocity and acceleration terms within theequations above are required to be known precisely. This does not meanthat number of sensors have to be increased, however. The requiredvelocities and accelerations can be generated from position signals ifprocessing speed is sufficiently high.

FIG. 4 shows the development of the required velocity and accelerationterms. Here, it is seen that the second converted signal 222 (Θ) issubjected to a digital low-pass filter 410 and then to a differentiationstage 420, to produce a discrete-time sampled velocity signal 460(Θ_Dot). The differentiation stage 420 includes a unit delay 422, asummer 424, and a gain element 426. The gain element 426 has a gain of1/T, where T is the sampling interval. In an example, the samplinginterval is 1 millisecond. The low-pass filter 410 is provided to removehigh-frequency content from the sampled motor shaft position. In anexample, the low-pass filter 410 has a cut-off frequency of 30 Hz andhas a transfer function of num(z)/(z−0.1518).

The first converted signal 212 (Θ_(d)) is subjected to a differentiationstage 430, to produce a discrete-time sampled velocity signal (Θ_(d)_(—) Dot). The differentiation stage 430 includes a unit delay 432, asummer 434, and a gain element 436. A summer 450 subtracts Θ_(d) _(—)Dot from Θ_Dot to produce a discrete-time sampled velocity differencesignal 470 (Diff_Dot). Also, Θ_(d) _(—) Dot is subjected to anotherdifferentiation stage 440, including elements 442, 444, and 446, toproduce a discrete-time sampled acceleration signal 480 (Θ_(d) _(—)Dot_Dot).

FIG. 5 shows an example realization of the sliding mode controller 240.Here, the sliding mode controller 240 is seen to receive the velocityand acceleration terms, whose generation is shown in FIG. 4, as well asthe difference signal 236 (FIG. 2). The sliding mode controller alsoreceives as input the four control gains (M_(c), η, λ and Φ) specifiedby the equations above. These control gains can be tuned as appropriateto the implementation. M_(c) can be pre-determined explicitly, whereasη, λ, and Φ can be adjusted by intentionally trading off robustness,bandwidth, and transient stability. In one example, these four valuescan be provided as constants. In other examples, the values of some orall of them can be varied to adapt the sliding mode controller 240 todifferent circumstances. In the example shown, only a single output isprovided, i.e., the sliding mode control signal 242.

The illustrated components of FIG. 5 provide a circuit realization ofEquations (9), (14) and (17). The sliding function “s” from Equation (9)is realized as the signal 418. The control law u_(SMC) from Equation(14) is realized as the signal 562. Also, the modified control lawu′_(SMC) from Equation (17), which is selected for operation within theboundary layer of thickness Φ, is realized as the signal 572.

The sliding function 418 (“s”) is developed in accordance with Equation(9) using a gain element 510, an integrator 512, a gain element 514, anda summer 516, configured in the manner shown. A signal 526 is common toboth u_(SMC) and u′_(SMC) and is developed in the manner shown using thea gain element 520, a gain element 522, and a summer 524. Also, a signal536 is developed in the manner shown using an absolute value element530, a multiplier 532, and a summer 534. In an example, the integrator512 has a transfer function KT(z+1)/[2(z−1)], where K is an adjustableconstant, such as 1.0, for example. The integrator 512 is provided todrive to zero steady-state errors caused by torque disturbances fromcoupled load-side structures.

The control law u_(SMC) (the signal 562) is then developed in the mannershown from the signal 418 (“s”), the signal 526, and the signal 536using a sign element 540, a multiplier 550, and a summer 560. Similarly,the modified control law u′_(SMC) (the signal 572) is developed in themanner shown from the signal 418 (“s”) and the signal 526 using a gainelement 546 and a summer 570.

A selector 580 selects between u_(SMC) (562) and u′_(SMC) (572) bycomparing the absolute value of “s” (418), obtained via an absolutevalue element 542, with the a boundary layer threshold 582. In anexample, the boundary layer threshold 582 is set to the boundary layerthickness, Φ. In the manner described in connection with Equation (17),the selector 580 provides u_(SMC) as its output when the absolute valueof “s” is greater than or equal to the boundary layer threshold 582 andprovides u′_(SMC) as its output when the absolute value of “s” is lessthan the boundary layer threshold 582. A proper value of Φ, and thus ofthe boundary layer threshold 582, may be established by trial and error.In an example, the value of Φ is set at between 1% and 5% the full-scalerange of u_(SMC). However, value of Φ is not necessarily confined to aconstant but can be made an adjustable parameter to adapt to dynamicmodel uncertainties over time.

With sliding mode operation thus established, a gain element 590 isprovided to adjust the output of the selector 580. In this example, thegain value G of the gain element 590 is set to R_(M)J_(M)/K_(T), whereR_(M), J_(M), and K_(T) are modeled characteristics of the brushless DCmotor 143. To explain, it is observed that the voltage V_(M) applied tothe brushless DC motor 142 is equal to the product of the voltage of thebattery 132, i.e., V_(SRC), times the control effort, PWM of thepulsewidth modulator 120, which is typically a fraction between 0 and 1.Stated mathematically,

V _(M) =PWM×V _(SRC).  (18)

Substituting EQ. (5B) into EQ. (18) then provides,

$\begin{matrix}{u_{SMC} = {{V_{SRC} \cdot {PWM} \cdot \frac{K_{T}}{R_{M}J_{M}}} = {{V_{SRC} \cdot {PWM} \cdot 1}\text{/}G}}} & (19)\end{matrix}$

Thus, it is seen that u_(SMC) inherently includes the factorK_(T)/R_(M)J_(M). It is evident from continuity arguments that the sameis true of u′_(SMC). Setting the gain G of the gain block 590 toR_(M)J_(M)/K_(T) thus cancels out the 1/G factor and corrects forparameters of the brushless DC motor 142. With this correction, thesliding mode control signal 242 is inherently left with a term that issignificantly affected by V_(SRC), the voltage of the battery 132.Referring briefly back to FIG. 2, it is seen that the source voltagenormalization circuit 270 divides the sliding mode control signal 242 byV_(SRC) (when the selector 260 selects the sliding mode control signal242), thus producing the control signal 116 in a form that does notdepend on V_(SRC), i.e., in a form that is insensitive to changes inV_(SRC).

The arrangement of FIG. 5 provides sliding mode control in accordancewith design requirements while reducing or eliminating chatter at thesliding surface. As the sliding surface, by definition, is where s=0,changing the gain of the sliding mode controller 240 within the boundarylayer that surrounds the sliding surface provides continuity of gainwithin the boundary layer, and thus avoids the discontinuity that wouldotherwise arise from the operation of the sign element 540.

FIG. 6 shows a phase plot of a dynamic response of the slidingcontroller 240 in the absence of the boundary layer, such that theselector 580 always selects the signal 562. Here, x represents theposition of the shaft 144 of the brushless DC motor 142, and {dot over(x)} represents the velocity of the shaft 144. The line 610 representsthe sliding surface, i.e., a line of the sliding function where s=0. Thecurve 612 represents the actual response of the motor shaft 144 whencontrolled by the sliding controller 240. Here it is seen that the curve612 quickly converges to the sliding surface 610, and proceeds tochatter as the response follows the sliding surface 610 to the desiredvalue, x_(d)(t). Chattering is caused by the operation of the SGNfunction of the sign element 540. As is known, the SGN function providesan output of one if its input is positive, an output of negative one ifits input is negative, and an output of zero if its input is zero. Thesystem trajectory 612 tends to overshoot the sliding surface 610,causing control to be tossed back-and-forth across the sliding surface610, because the feedback gain of the sliding controller 240 alternatesin sign (on account of the SGN function) as the value of “s” crossesbetween positive and negative values. Lines 620 represent the gaingradient for s>0, whereas lines 630 represent the gain gradient for s<0.In other words, the discontinuity in gain, between a first value for s>0and a second value s<0, contributes to the chattering effect.

FIG. 7 shows a phase plot for the sliding controller 240 where the widthΦ of the boundary layer is greater than zero. Here, a boundary layer 720is shown around the sliding surface 610. Unlike the case in FIG. 6, thefeedback gain in this example is continuous in the vicinity of thesliding surface 610, reducing or eliminating the effects of chatter.

FIG. 8 shows an example implementation of the second controller 250.Here, the second controller 250 takes the form of a PID controllerhaving a first gain element 810, a PID stage 820, and a second gainelement 830. The second gain element 830 is seen to have a gain ofV_(Nom), which represents a nominal voltage of the battery 132. Thesecond gain element 830 thus operates in cooperation with the sourcevoltage normalization circuit 270 to compensate for changes in batteryvoltage. In particular, the gain element 830 together with the sourcevoltage normalization circuit 270 provides unity gain when the signal136 reports a battery voltage equal to V_(Nom).

Although FIG. 8 shows the second controller 250 implemented as a PIDcontroller, this is merely an example. Alternatively, the secondcontroller 250 may be realized with a Kalman controller, an H2controller, or an H-infinity controller and so on, for example.

FIG. 9 illustrates a process 900 that may be carried out in connectionwith the apparatus 100. This process is typically performed by theconstructs described in connection with the control circuit 110 as shownin FIGS. 2-5 and 8. The various acts of the process 900 may be orderedin any suitable way. Accordingly, embodiments may be constructed inwhich acts are performed in orders different from those illustrated,which may include performing some acts simultaneously, even though theacts are shown as sequential in the illustrated embodiments.

At step 910, a first signal is received that indicates a desiredposition of an electro-mechanical actuator. For example, the controlcircuit 110 receives the signal 112 indicating the desired position ofthe motor shaft 144. The signal 112 may be provided in any suitableunits, such as degrees, for example.

At step 912, a second signal is received that indicates an actualposition of the electro-mechanical actuator. For example, the controlcircuit 110 receives the signal 148 indicating the rotational positionof the motor shaft 144 as measured by the sensor 146. In an alternativearrangement, a position of the flap 150 can be measured directly if asuitable sensor is provided.

At step 914, an error signal is calculated based on the first signal andthe second signal. For example, the error circuit 230 calculates thesignal 238 (E_(Norm)) based on the signal 112 and the signal 148, in themanner described in connection with FIG. 2.

At step 916, the position of the electro-mechanical actuator iscontrolled in a sliding control mode when the error signal is above apredetermined threshold. For example, as shown in FIG. 2, the selector260 selects the sliding mode control signal 242 for its output when thesignal 238 (E_(Norm)) exceeds the threshold 114.

At step 918, the position of the electro-mechanical actuator iscontrolled in a second control mode when the error signal is below thepredetermined threshold. For example, as shown in FIG. 2, the selector260 selects the second control signal 252 for its output when the signal238 (E_(Norm)) is less than the threshold 114.

An improved technique has been described for controlling anelectro-mechanical actuator. The technique combines a sliding mode ofcontrol with a second mode of control. An error signal is generatedbased on the difference between an input position signal and a feedbackposition signal. When the error signal is above a predeterminedthreshold, the actuator is controlled in the sliding control mode. Whenthe error signal is below the predetermined threshold, the actuator iscontrolled in the second control mode. The combination of the slidingcontrol mode with the second control mode yields a robust controllerthat can tolerate large parameter variations and uncertainties withoutsacrificing precise steady state tracking.

As used throughout this document, the words “comprising,” “including,”and “having” are intended to set forth certain items, steps, elements,or aspects of something in an open-ended fashion. Although certainembodiments are disclosed herein, it is understood that these areprovided by way of example only and the invention is not limited tothese particular embodiments.

Having described certain embodiments, numerous alternative embodimentsor variations can be made. For example, although the improved techniquehas been described for use in airborne applications, it can also beapplied in a myriad of other applications for controlling the positionof electro-mechanical actuators.

Also, although improvements have been described in connection withbrushless DC motors, they may be applied with other types of motors,such as stepper motors, for example.

Also, although a particular combination of sliding mode control and asecond mode of control are described for controlling the position of anactuator, embodiments of the invention may alternatively be expressed asa combination of a first sliding mode control, which includes anintegral term, and a second sliding mode of control, which includesboundary layer control. A transition is made between the first slidingmode control and the second sliding mode control in the vicinity of thesliding surface. Secondarily, an additional control mode may be applied,such as PID control or some other type of control that is notsusceptible to chattering, and such additional control mode may beswitched in for controlling the actuator in the vicinity of a steadystate value, e.g., when the position of the actuator approaches itsprogrammed position.

Further, although specific features are shown and described withreference to particular embodiments, such features may be included inany of the disclosed embodiments and their variants. Thus, it isunderstood that features disclosed in connection with any embodiment areincluded as variants of any other embodiment, whether such inclusion isexplicit or not.

Further still, the improvement or portions thereof may be embodied as anon-transient computer-readable storage medium, such as a magnetic disk,magnetic tape, compact disk, DVD, optical disk, flash memory, and thelike (shown by way of example as medium 950 in FIG. 9). Multiplecomputer-readable media may be used. The medium (or media) may beencoded with instructions which, when executed on one or moreprocessors, perform methods that implement the various processesdescribed herein. Such medium (or media) may be considered an article ofmanufacture or a machine, and may be transportable from one machine toanother.

Those skilled in the art will therefore understand that various changesin form and detail may be made to the embodiments disclosed hereinwithout departing from the scope of the invention.

What is claimed is:
 1. A method of controlling an electro-mechanical actuator, comprising: receiving a first signal indicating a desired position of the electro-mechanical actuator; receiving a second signal indicating an actual position of the electro-mechanical actuator; calculating an error signal based on the first signal and the second signal; controlling the position of the electro-mechanical actuator in a sliding control mode when the error signal is above a predetermined threshold; and controlling the position of the electro-mechanical actuator in a second control mode when the error signal is below the predetermined threshold.
 2. The method of claim 1, wherein the second control mode is selected from the group consisting of (i) a proportional-integral-derivative (PID) control mode, (ii) a Kalman control mode, (iii) an H2 control mode, and (iv) an H-infinity control mode.
 3. The method of claim 2, wherein controlling the position of the electro-mechanical actuator in the sliding control mode includes: computing a value of a sliding function s such that s=0 defines a sliding surface of the sliding mode control; generating a sliding mode control signal via a first path when the computed value of s falls outside a boundary layer around the sliding surface; and generating the sliding mode control signal via a second path when the computed value of s falls within the boundary layer around the sliding surface.
 4. The method of claim 3, wherein the first path has a first gain on one side of the sliding surface (s<0) and a second gain on another side of the sliding surface (s>0), wherein the second path provides a transition between the first gain and the second gain within the boundary layer.
 5. The method of claim 4, wherein the transition between the first gain and the second gain provided by the second path within the boundary layer is substantially linear.
 6. The method of claim 4, wherein controlling the position of the electro-mechanical actuator in the sliding control mode further includes receiving an error signal proportional to the difference between the first signal and the second signal and integrating the error signal.
 7. The method of claim 4, wherein controlling the position of the electro-mechanical actuator in the sliding control mode further includes: driving a pulsewidth modulator with the sliding mode control signal to produce a pulsewidth modulated signal; driving a brushless DC motor with a power source switched to the brushless DC motor using the pulsewidth modulated signal; and driving the actuator with the brushless DC motor.
 8. The method of claim 7, further comprising establishing control parameters of the sliding control mode based on modeled physical characteristics of the brushless DC motor.
 9. The method of claim 8, further comprising: measuring a voltage of the power source; and dividing the sliding mode control signal by the measured voltage to compensate for variations in the voltage of the power source.
 10. The method of claim 7, wherein controlling the position of the electro-mechanical actuator in the second control mode includes: generating a second control signal; driving the pulsewidth modulator with the second control signal to produce the pulsewidth modulated signal; driving the brushless DC motor with a power source switched to the brushless DC motor using the pulsewidth modulated signal; and driving the actuator with the brushless DC motor.
 11. The method of claim 10, further comprising: measuring a voltage of the power source; and normalizing the second control signal based on the measured voltage of the power source to compensate for variations in the voltage of the power source.
 12. The method of claim 7, wherein calculating the error signal includes: calculating a difference signal proportional to a difference between the first signal and the second signal; calculating a magnitude of the quotient of the difference signal divided by the first signal.
 13. A control circuit for controlling an electro-mechanical actuator, comprising: a sliding mode controller configured to generate a sliding mode control signal; a second controller configured to generate a second mode control signal; an error circuit configured to generate an error signal based on a difference between an input signal indicative of a desired position of the electro-mechanical actuator and a feedback signal indicative of an actual position of the electro-mechanical actuator; and a selector circuit coupled to the sliding mode controller, the second controller, and the error circuit, and configured (i) to select the sliding mode control signal to control the electro-mechanical actuator when the error signal is above a predetermined threshold and (ii) to select the second mode control signal to control the electro-mechanical actuator when the error signal is below the predetermined threshold.
 14. The control circuit of claim 13, wherein the second controller is selected from the group consisting of (i) a proportional-integral-derivative (PID) controller, (ii) a Kalman controller, (iii) an H2 controller, and (iv) an H-infinity controller.
 15. The control circuit of claim 14, wherein, when configured to generate the sliding mode control signal, the sliding mode controller is further configured to: compute a sliding function s such that s=0 defines a sliding surface of the sliding mode control; generate the sliding mode control signal via a first path when s falls outside a boundary layer around the sliding surface; and generate the sliding mode control signal via a second path when s falls within the boundary layer around the sliding surface.
 16. The control circuit of claim 15, wherein the first path has a first gain on one side of the sliding surface (s<0) and a second gain on another side of the sliding surface (s>0), wherein the second path provides a substantially linear transition between the first gain and the second gain within the boundary layer.
 17. The control circuit of claim 16, wherein the error circuit, when configured to generate the error signal, is further configured to generate a difference signal proportional to a difference between an input signal indicative of a desired position of the electro-mechanical actuator and a feedback signal indicative of an actual position of the electro-mechanical actuator, and wherein the sliding mode controller includes an integrator configured to integrate the difference signal.
 18. The control circuit of claim 17, wherein the selector has an output coupled to a motor for driving the electro-mechanical actuator, wherein the motor is driven from a power source, wherein a measurement circuit measures a voltage of the power source, and wherein the control circuit further comprises a dividing circuit, coupled to the output of the selector and to the measurement circuit, and configured to divide the selected control signal by the measured voltage of the power source.
 19. A non-transitory computer-readable medium including instructions which, when executed by a control circuit, cause the control circuit to perform a method for controlling an electro-mechanical actuator, the method comprising: receiving a first signal indicating a desired position of the electro-mechanical actuator; receiving a second signal indicating an actual position of the electro-mechanical actuator; calculating an error signal based on the first signal and the second signal; controlling the position of the electro-mechanical actuator in a sliding control mode when the error signal is above a predetermined threshold; and controlling the position of the electro-mechanical actuator in a second control mode when the error signal is below the predetermined threshold.
 20. The non-transient computer-readable medium of claim 19, wherein controlling the position of the electro-mechanical actuator in the sliding control mode includes: computing a sliding function s such that s=0 defines a sliding surface of the sliding mode control; generating a sliding mode control signal via a first path when s falls outside a boundary layer around the sliding surface; and generating the sliding mode control signal via a second path when s falls within the boundary layer around the sliding surface, wherein the first path has a first gain on one side of the sliding surface (s<0) and a second gain on another side of the sliding surface (s>0), wherein the second path provides a substantially linear transition between the first gain and the second gain within the boundary layer. 